The maximum independent union of cliques problem: complexity and exact approaches

Ertem, Zeynep; Lykhovyd, Eugene; Wang, Yiming; Butenko, Sergiy

doi:10.1007/s10898-018-0694-2

Cited by 13 publications

(12 citation statements)

References 33 publications

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“…(ii) Exploring various combinatorial objects representing groups of similar materials in the constructed unweighted materials networks as well as their edge-weighted counterparts, with the edge weights given by the computed similarity scores. Edge-weighted cliques [34][35][36] and clique relaxation models 37,38 are of particular interest in this regard. (iii) Utilizing the proposed methodology in a context of specific applications, such as determining suitable materials for manufacturing composites with desired properties.…”

Section: Discussionmentioning

confidence: 99%

Networks of materials: Construction and structural analysis

et al. 2020

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Modeling and analysis of the materials universe is an emerging area of research with many important applications in materials science. The main goal is to create a map of materials which allows not only to visualize and navigate the materials space, but also reveal complex relationships and "connections" among materials and potentially find clusters of materials with similar properties. In this paper, we consider the problem of mapping and exploring the materials universe using network science tools and concepts. The networks are based on the open-source materials data repository AFLOW.org where each material is represented as a node, and each pair of nodes is connected by a link if the respective materials exhibit a high level of similarity between their Density of States (DOS) functions. We discuss the importance of similarity measure selection, investigate basic structural properties of the resulting networks, and demonstrate advantages and limitations of the proposed approaches. Materials networks, similarity measures, DOS function, materials informatics, network analysis, clique.

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Section: Discussionmentioning

confidence: 99%

Networks of materials: Construction and structural analysis

et al. 2020

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“…However, the LP relaxation of ( 5) is generally too weak. In fact, given the feasibility of the fractional assignment x i = 2 3 , ∀i ∈ V , the optimal solution value of the LP relaxation problem is at least as large as 2n 3 , while the computational results of Ertem et al [19] show that the IUC number of graphs with moderate densities tends to stay in a close range of their relatively small independence and clique numbers. This is due to the fact that the fractional IUC polytope…”

Section: The Iuc Polytopementioning

confidence: 99%

“…Theorem 8 is closely related to a basic property of IUCs. Ertem et al [19] showed that, in an arbitrary graph, a (maximal) clique is a maximal IUC if and only if it is a dominating set. In an anti-cycle graph G = (A, E), each maximal clique is maximum, and in case |A| ≥ 6, it is also a maximum IUC by Theorem 8.…”

Section: Chordless Cycle and Its Complementmentioning

confidence: 99%

“…They proposed a graph clustering algorithm based on a given maximum IUC, referred to as disjoint 1-clusters in [20], and showed the competence of their method via experiments with real-life social networks. It was not until very recently that the term "independent union of cliques" was used by Ertem et al [19]. In this work, the authors studied some basic properties of IUCs and analyzed complexity of the Maximum IUC problem on some restricted classes of graphs.…”

Section: Introductionmentioning

confidence: 99%

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Polyhedral Properties of the Induced Cluster Subgraphs

Hosseinian¹,

Butenko²

2019

Preprint

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A cluster graph is a graph whose every connected component is a complete graph. Given a simple undirected graph G, a subset of vertices inducing a cluster graph is called an independent union of cliques (IUC), and the IUC polytope associated with G is defined as the convex hull of the incidence vectors of all IUCs in the graph. The Maximum IUC problem, which is to find a maximum-cardinality IUC in a graph, finds applications in network-based data analysis. In this paper, we derive several families of facet-defining valid inequalities for the IUC polytope. We also give a complete description of this polytope for some special classes of graphs. We establish computational complexity of the separation problem for most of the considered families of valid inequalities and explore the effectiveness of employing the corresponding cutting planes in an integer (linear) programming framework for the Maximum IUC problem through computational experiments.

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“…holds for any graph G. However, the disjoint union of k ≥ 1 cliques of size k provides an example on k 2 vertices where α ω (G) = k 2 is much greater than both the independence number α(G) = k and the clique number ω(G) = k. In the other direction, it is obvious that α ω (G) ≤ α(G) ω(G). This parameter was first introduced in this form by Ertem et al [14], but it can be shown to be equivalent to the Cluster Vertex Deletion problem [7]. There is a large volume of literature on the computational aspects of related parameters, see for example [15].…”

Section: Introductionmentioning

confidence: 99%

Independent Chains in Acyclic Posets

Salia,

Spiegel,

Tompkins

et al. 2019

Preprint

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We consider the problem of determining the maximum order of an induced vertex-disjoint union of cliques in a graph. More specifically, given some family of graphs G of equal order, we are interested in the parameterWe determine the value of this parameter precisely when G is the family of comparability graphs of n-element posets with acyclic cover graph. In particular, we show that α ω (G) = (n + o(n))/ log 2 (n) in this class.

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The maximum independent union of cliques problem: complexity and exact approaches

Cited by 13 publications

References 33 publications

Networks of materials: Construction and structural analysis

Networks of materials: Construction and structural analysis

Polyhedral Properties of the Induced Cluster Subgraphs

Independent Chains in Acyclic Posets

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